Question

1. # Find An Equation In Standard Form For The Hyperbola With Vertices At (0, ±9) And Foci At (0, ±10).

The standard form of equation for a hyperbola is an equation that is used to represent the shape of a hyperbola on the x-y coordinate plane. This equation can also be used to determine certain properties of the hyperbola, such as its vertices and foci. In this blog post, we will discuss how to find an equation in standard form for the hyperbola with vertices at (0, ±9) and foci at (0, ±10). We will also explore how this equation can be used to calculate the lengths of the transverse and conjugate axes of the hyperbola.

## What is a hyperbola?

A hyperbola is a type of curve that results from the intersection of a cone and a plane that is not parallel to the base of the cone. The standard equation for a hyperbola with vertices at (, ±) and foci at (, ±) can be written as:

\$\$frac{(x-h)^2}{a^2} – frac{(y-k)^2}{b^2} = 1\$\$

where \$h\$ and \$k\$ are the coordinates of the center of the hyperbola, and \$a\$ and \$b\$ are its semi-major and semi-minor axes, respectively.

## What is standard form?

Standard form is a way of writing down equations that makes them easier to read and understand. In standard form, an equation will always have the same number of terms on each side of the equals sign. For example, the equation x + 3 = 5 would be written in standard form as x = 2.

Standard form also allows you to see what the different parts of an equation represent. For example, in the equation x + 3 = 5, the term x represents the unknown value that you are solving for, while the terms 3 and 5 represent known values. The term 3 is called a constant, while the term 5 is called a coefficient.

## How to find the equation for a hyperbola with given vertices and foci

To find the equation for a hyperbola with given vertices and foci, use the following steps:

1. Find the center of the hyperbola using the midpoint formula. The center is located at (h, k).

2. Find the length of the focal chord using the distance formula. The focal chord has length 2a.

3. Use the information from steps 1 and 2 to write the equation for a hyperbola in standard form.

The equation for a hyperbola in standard form is (x-h)^2/(a^2) – (y-k)^2/(b^2) = 1, where (h,k) is the center of the hyperbola, 2a is the length of the focal chord, and b is the distance from the vertex to focus.

## Examples of finding equations for hyperbolas in standard form

A hyperbola is a type of conic section that has two halves, each of which is a mirror image of the other. The line that divides the two halves is called the axis of symmetry. A hyperbola can be in either standard form or vertex form.

To find an equation for a hyperbola in standard form, you will need to know the coordinates of the vertices and the foci. The vertices are the points where the two halves of the hyperbola meet. The foci are the points on the axis of symmetry where the curve bends inward.

For example, consider the hyperbola with vertices at (2, 3) and (-1, -3), and with foci at (0, 0) and (4, 0). The equation for this hyperbola in standard form is:

(x-h)^2/(a^2) – (y-k)^2/(b^2) = 1

where h and k are the coordinates of the center of the hyperbola, and a and b are its semi-major and semi-minor axes respectively. In this case, h = 2 and k = 3. To find a and b, we can use either point on the axis of symmetry and one of the foci:

(0-2)^2/(a^2) – (0-3)^2/(b^2

## Conclusion

We have found an equation in standard form for a hyperbola with vertices at (0, ±9) and foci at (0, ±10). This hyperbola is defined by the equation x^2/81 – y^2/100 = 1. Being able to understand how to solve equations like this one can be useful when studying mathematics or physics as it could help you better visualize problems and gain a deeper understanding of them. Additionally, being able to recognize the shape of a hyperbola from its equation is also important as it will allow you to quickly identify them in future questions.

2. Do you need to find an equation in standard form for a hyperbola with vertices at (0, ±9) and foci at (0, ±10)?

No worries! You’ve come to the right place.

There are a few steps to finding the equation in standard form for a hyperbola with vertices at (0, ±9) and foci at (0, ±10). Let’s walk through them together!

First, let’s calculate the length of the major axis, which is the distance between the vertices. This is done by subtracting the y-coordinates of the vertices. In this case, the major axis is 18.

Next, let’s calculate the length of the minor axis, which is the distance between the foci. This is done by subtracting the y-coordinates of the foci. In this case, the minor axis is 2.

Now, let’s calculate the distance between the center of the hyperbola and either focus (the distance between the vertex and the focus). This is done by subtracting the y-coordinates of the vertex and the focus. In this case, the distance is 1.

Finally, let’s put all of this information together to create the equation in standard form for the hyperbola. The equation is:

((x – 0)²/18²) – ((y – 9)²/1²) = 1.

There you have it! With a few simple steps, you’ve found the equation in standard form for the hyperbola with vertices at (0, ±9) and foci at (0, ±10). Now you’re ready to take on any hyperbola challenge that comes your way!